Related papers: Gaussian Beam Methods for the Helmholtz Equation
In this paper we present and analyse a high accuracy method for computing wave directions defined in the geometrical optics ansatz of Helmholtz equation with variable wave number. Then we define an "adaptive" plane wave space with small…
We study the approximability of general convex sets in $\mathbb{R}^n$ by intersections of halfspaces, where the approximation quality is measured with respect to the standard Gaussian distribution $N(0,I_n)$ and the complexity of an…
In this paper, a generalized finite element method (GFEM) with optimal local approximation spaces for solving high-frequency heterogeneous Helmholtz problems is systematically studied. The local spaces are built from selected eigenvectors…
Propagation of high-frequency wave in periodic media is a challenging problem due to the existence of multiscale characterized by short wavelength, small lattice constant and large physical domain size. Conventional computational methods…
This paper develops and analyzes some interior penalty discontinuous Galerkin methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in the two and three dimensions. It is…
Exact Bateman-Hillion solutions of the wave equation are applied to Hermite-Gaussian beams using a space-time constraint condition that requires the field density to fall as the inverse square of distance from the focal point of the beam at…
A concise method of deriving expressions for Gaussian-like solutions of the paraxial and d'Alembert equations is presented. This method is based on the Hankel transform. Choosing some Gaussian base functions with slight modifications of the…
We deduce a set of circularly polarized Gaussian laser beam modes via a separation-of-variables solution to the Helmholtz wave equation in oblate spheriodal coordinates. On transforming to cylindrical coordinates these become the well-known…
We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when…
In this paper, we propose a method to approximate the Gaussian function on ${\mathbb R}$ by a short cosine sum. We generalise and extend the differential approximation method proposed in [4, 40] to approximate $\mathrm{e}^{-t^{2}/2\sigma}$…
For $h$-FEM discretisations of the Helmholtz equation with wavenumber $k$, we obtain $k$-explicit analogues of the classic local FEM error bounds of [Nitsche, Schatz 1974], [Wahlbin 1991], [Demlow, Guzm\'an, Schatz 2011], showing that these…
In this paper, the generalized finite element method (GFEM) for solving second order elliptic equations with rough coefficients is studied. New optimal local approximation spaces for GFEMs based on local eigenvalue problems involving a…
We consider Friedlander's wave equation in two space dimensions in the half-space x > 0 with the boundary condition u(x,y,t)=0 when x=0. For a Gaussian beam w(x,y,t;k) concentrated on a ray path that is tangent to x=0 at (x,y,t)=(0,0,0) we…
We consider the light scattering problem for a Gaussian beam and a (spherical) particle at arbitrary location. Within the beam cross section, the total electromagnetic field is the superposition of the incident beam and the scattered wave.…
We consider the problem of approximating a function using Herglotz wave functions, which are a superposition of plane waves. When the discrepancy is measured in a ball, we show that the problem can essentially be solved by considering the…
In this work it is shown how to obtain, in a simple way, localized (non- diffractive) subluminal pulses as exact analytic solutions to the wave equations. These new ideal subluminal solutions, which propagate without distortion in any…
For the $h$-finite-element method ($h$-FEM) applied to the Helmholtz equation, the question of how quickly the meshwidth $h$ must decrease with the frequency $k$ to maintain accuracy as $k$ increases has been studied since the mid 80's.…
It is often convenient to use Gaussian blur in studying image quality or in data augmentation pipelines for training convoluional neural networks. Because of their convenience, Guassians are sometimes used as first order approximations of…
In this work we consider the computational approximation of a unique continuation problem for the Helmholtz equation using a stabilized finite element method. First conditional stability estimates are derived for which, under a convexity…
Usual Gaussian beams are particular scalar solutions to the paraxial Helmholtz equation, which neglect the vector nature of light. In order to overcome this inconvenience, Simon et al. (J. Opt. Soc. Am. A 1986, 3, 536-540) found a paraxial…